Targonskii, A. L.

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А.L. TargonskiiExtremal problems for partially non-overlapping domains onequiangular system points.Let N, R – the sets natural and real numbers conformity, C – the plaincomplex numbers, C = C⋃{∞} – the Riemannian sphere, R+ = (0,∞).For fix number n ∈ N system pointsAn = {ak ∈ C : k = 1, n},the define n-equiangular system points, if by all k = 1, n realize relation:arg ak =2pin(k − 1), k = 1, n. (1)System be considered the angular domains:Pk = {w ∈ C : 2pin(k − 1) < argw < 2pink}, k = 1, n.For arbitrary n-equiangular system points "controlling" functional to beconsideredµ (An) :=n∏k=1χ(∣∣∣∣ak+1ak∣∣∣∣n4)· |ak|,where χ(t) = 12(t+ 1t), t ∈ R+.Let D, D ⊂ C – the arbitrary open set and w = a ∈ D, this D(a)the define connected component D, the contain point a. For arbitrary n-equiangular system points An = {ak}nk=1 and open set D, An ⊂ D the defineDk(ap) connected component set D(ap)⋂Pk, the contain point ap, k = 1, n,p = k, k + 1, s = 1,m, an+1 := a1. Let Dk(0) (conformity Dk(∞)) the defineconnected component set D(0)⋂Pk (conformity D(∞)⋂Pk), the containpoint w = 0 (conformity w =∞).The define, what open set D, {0,∞}∪An ⊂ D satisfy the conditions non-overlapping relatively n-equiangular system points An if be satisfied condition[Dk(ak)⋂Dk(ak+1)]⋃[Dk(0)⋂Dk(ak)]⋃[Dk(0)⋂Dk(∞)]⋃⋃[Dk(∞)⋂Dk(ak)]⋃[Dk(∞)⋂Dk(ak+1)]⋃[Dk(0)⋂Dk(ak+1)]= ∅,(2)1k = 1, n on all angular domains Pk.System domains {Bk}nk=1, k = 1, n, the define system partially non-overlapping domains, ifD :=n⋃k=1Bk, (3)is open sets, the satisfied condition (2).Let r(B; a) – inner radius domain B ⊂ C with respect to a point a ∈ B.Theorem. Let γ ∈ R+, n ∈ N, n ≥ 3. Then for arbitrary n-equiangularsystem points (1), the satisfied conditionµ (An) = 1,and arbitrary set partially non-overlapping domains {B0, Bk, B∞}, the satisfiedcondition (3), ak ∈ Bk ⊂ C, k = 1, n, 0 ∈ B0 ⊂ C, ∞ ∈ B∞ ⊂ C, be satisfiedinequality(r (B0 ; 0) · r (B∞ ; ∞))γ ·n∏k=1r (Bk ; ak) ≤≤ (r (B00 ; 0) · r (B0∞ ; ∞))γ · n∏k=1r(B0k ; a0k).The equality obtain in this inequality, when points {a0k} and domains {B00 , B0k, B0∞},k = 1, n are, conformity, the poles and the circular domains of the quadraticdifferentialQ(w)dw2 = −γw2n + (n2 − 2γ)wn + γw2 (wn − 1)2 dw2.2

Extremal problems for partially non-overlapping domains on equiangular system points

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